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Mirrors > Home > MPE Home > Th. List > f1orn | Structured version Visualization version GIF version |
Description: A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.) |
Ref | Expression |
---|---|
f1orn | ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff1o2 6854 | . 2 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹)) | |
2 | eqid 2735 | . . 3 ⊢ ran 𝐹 = ran 𝐹 | |
3 | df-3an 1088 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹) ∧ ran 𝐹 = ran 𝐹)) | |
4 | 2, 3 | mpbiran2 710 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹)) |
5 | 1, 4 | bitri 275 | 1 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ◡ccnv 5688 ran crn 5690 Fun wfun 6557 Fn wfn 6558 –1-1-onto→wf1o 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-ex 1777 df-cleq 2727 df-ss 3980 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 |
This theorem is referenced by: f1f1orn 6860 infdifsn 9695 efopnlem2 26714 cycpmcl 33119 |
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